sidef-scripts/Math/ramanujan_sum.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 26 July 2017
# https://github.com/trizen

# Ramanujan's sum:
#   c_k(n) = Sum_{m mod k; gcd(m, k) = 1} exp(2*pi*i*m*n/k)

# For n = 1, c_k(1) is equivalent to moebius(k).

# For integer real values of `n` and `k`, Ramanujan's sum is equivalent to:
#   c_k(n) = Sum_{m mod k; gcd(m, k) = 1} cos(2*pi*m*n/k)

# Alternatively, when n = k, `c_n(n)` is equivalent with `euler_phi(n)`.

# The record values, `c_n(n) + 1`, are the prime numbers.

define tau_i = Num.tau.i

func ramanujan_sum(n, k) {
    sum(1..k, {|m|
        gcd(m, k) == 1 ? exp(tau_i * m * n / k)
                       : 0
    }).round(-40)
}

each(1..30, {|n|
    var r = ramanujan_sum(n, n**2)
    say "R(#{n}, #{n**2}) = #{r}"
})

__END__
R(1, 1) = 1
R(2, 4) = -2
R(3, 9) = -3
R(4, 16) = 0
R(5, 25) = -5
R(6, 36) = 6
R(7, 49) = -7
R(8, 64) = 0
R(9, 81) = 0
R(10, 100) = 10
R(11, 121) = -11
R(12, 144) = 0
R(13, 169) = -13
R(14, 196) = 14
R(15, 225) = 15
R(16, 256) = 0
R(17, 289) = -17
R(18, 324) = 0
R(19, 361) = -19
R(20, 400) = 0
R(21, 441) = 21
R(22, 484) = 22
R(23, 529) = -23
R(24, 576) = 0
R(25, 625) = 0
R(26, 676) = 26
R(27, 729) = 0
R(28, 784) = 0
R(29, 841) = -29
R(30, 900) = -30